Fast adiabatic pulses

ABSTRACT

New methods of generating optimal inversion pulses and adiabatic pulses in magnetic resonance imaging are disclosed. Trajectories, maximum sweep rates, and velocity profiles are used in defining an optimal pulse, over support regions. Adiabatic pulses are optimized by using the trajectory as a constraint of optimization, but selecting trajectories with velocity profiles, without using the adiabatic condition as a constraint for optimizing the velocity profile. A method or inverting MR spins substantially, independently of the pulse duration, by selecting a transition width between 1.4 and 1.9 and dividing that width by the pulse duration is disclosed; and A new method of inverting adiabatic, MR, amplitude modulated spins, with a trajectory defined by sin α/cos α, where α&lt;0.9 and at least 50% of the trajectory is outside, in a z-x rotating frame of reference that rotates at the instantaneous frequency of the RF pulse is also taught.

FIELD OF THE INVENTION

The present invention relates to the design of adiabatic pulses for MRI(Magnetic Resonance Imaging) and in particular to an adiabatic pulsehaving a substantially half-rectangle trajectory.

BACKGROUND OF THE INVENTION

Magnetic resonance imaging is based on the process of inverting thespins of atoms which are situated in a strong axial magnetic field andthen measuring the electromagnetic radiation of the atoms, as the spinsreturn to a more relaxed state. A practical MRI device requires theability to selectively invert a narrow slice of a subject, in a shortperiod of time and using a low dose of RF radiation. The usual manner ofinversion includes applying a z-gradient magnetic field to the subjectso that each slice of the subject has a different Larmor frequency andirradiating the subject with an RF radiation pulse, so that only thespins in one of these slices are inverted. As higher strength magneticfields are used for MRI imaging, the amount of RF energy absorbed by thebody is higher. It is therefore important to limit the amount ofradiation to which the subject is exposed. Further, in many MRI devices,the peak RF amplitude is limited. Usually, there is a tradeoff madebetween the pulse duration and the RF amplitude.

The relationship between the RF radiation, the magnetic field and theinversion of the spins is described by the Bloch equations, for whichthere are only a small number of known analytical solutions.

When an RF electromagnetic field is applied to a spin which is alreadyin a strong static magnetic field, the RF magnetic field affects thespin. The RF field is very much smaller than the static field, so the RFfield is usually described as rotating in the plane perpendicular to thefield direction of the static magnetic field (the effect of thecomponent in the static field direction is negligible). The effect ofthe RF field on the spins is most conveniently described in a rotatingframe of reference, having three perpendicular axes, Z, Y and X. The Zaxis is aligned with the main magnetic field denoted by M_(z). The Xaxis is aligned with the RF field and the Y axis is perpendicular toboth the X and Z axes. The entire frame of reference rotates around theZ axis at the instantaneous angular (frequency) of the RF pulse. Both Xand Z axes use units of frequency, such that all magnetic fields B arerepresented by vectors γB, where γ is the gyro-magnetic resonancecoefficient for the spin (type of species thereof).

The effective magnetic field to which a spin is subjected as a result ofthe RF field is preferably defined as a vector in the rotating frame ofreference. The magnitude of the Z component of the vector is equal tothe frequency difference between the RF field frequency and the Larmorfrequency of the spin. The magnitude of the X component is equal to theinstantaneous amplitude of the RF field. It should be appreciated thatin a uniform Z directed field, all the spins are located at the same Zcoordinate. When a gradient magnetic field is applied, each spin has adifferent Larmor frequency and, hence, a different Z coordinate.

Typically, the net magnetization of a group of spins is treated as asingle vector value, called the magnetization vector. Thus, the effectof an inversion pulse is to invert the magnetization vector in a sliceof tissue. FIG. 1 is a graph of a typical inverted slice profile inwhich a nomalized magnetization is shown as a function of anoff-resonance frequency. The slice includes an in-slice region, which isinverted by the inversion pulse, an out-of-slice region which is notinverted by the pulse and a transition region where the post-inversionmagnetization varies between +1 (not inverted) and -1 (inverted). Themagnetization values are normalized to the equilibrium magnetization,M₀. For convenience, the in-slice region is usually depicted as centeredaround the magnetization axis. The width of the slice (SW) is usuallymeasured between the two points where the post-inversion magnetizationvector is zero. The slice width is measured in units of frequency, whichreflect the relationship between the inversion and the Larmor frequency.For convenience, C_(o) is defined to be half the transition width.

One important type of inversion pulse is an adiabatic pulse. Inversionby adiabatic pulse is less affected by inhomogenities of the RF fieldamplitude than is inversion by other types of inversion pulses. Anadiabatic pulse uses the following mechanism: An effective magnetizationof the RF radiation field is initially aligned with the main fieldmagnetization axis (+M_(z)) direction and is slowly changed until it isaligned in the direction opposite the main field magnetization (-M_(z)).If the rate of change of the effective magnetization vector is gradualenough, the magnetization vector will track the effective magnetizationof the RF field and will be inverted when the effective magnetizationvector becomes aligned with the -Z axis. The adiabatic condition(described below) describes the conditions under which the rate ofchange of the vector is sufficiently gradual to permit tracking. Themotion of the effective magnetization is characterized by its"trajectory", which is the path of the tip of the effectivemagnetization vector and its "velocity profile", which describes theinstantaneous rate of motion of the effective magnetization vector alongits trajectory.

FIG. 2 is a graph showing the trajectory of a typical adiabatic pulse inthe Z-X plane. The effective magnetization vector of the pulse startsout aligned with the +Z direction and moves along a half ellipse in theZ-X plane until it becomes aligned with the -Z direction. It should benoted that the trajectory shown in FIG. 2 is only correct for spins atthe center of the slice. For all other spins, the shown trajectory isshifted by an amount equal to the difference between the Larmorfrequency of the spin and the Lartnor frequency at the slice center. Foreach point P along the trajectory, which indicates an instantaneousposition of the effective magnetization vector, x is the instantaneousRF amplitude and z is the instantaneous RF synthesizer frequency. Foreach spin which is affected by the adiabatic pulse, a vector connectingthe spin and point P is the effective field vector, having a magnituder. θ is defined as the angle between r and the X axis. In order for therate of change of the vector to be sufficiently gradual to permittracking, the motion must satisfy the following (adiabatic) condition,Γ=r/|θ|>>1, where Γ is an adiabatic parameter which describes thisratio. For the same magnetization vector traversing a given trajectoryat a given rate of motion, different spins will see different angularvelocities. Since r and θ are different for each spin, the adiabaticparameter may ensure tracking for one group of spins but not foranother, even at the same point P.

As can be appreciated, if θ is larger, the pulse will be shorter,however, the adiabatic parameter will be smaller, so tracking may breakdown and not be possible. In some MRI imaging sequences, time is ofessence, so a short inversion pulse is desired.

One of the most efficient (fast, low peak RF amplitude and adiabatic fora wide range of RF amplitudes) inversion pluses in the prior art is thesech/tanh pulse. The first term (sech) defines the X component of themagnetization vector and the second term (tanh) describes the Zcomponent. The trajectory of the sech/tanh pulse is a half ellipse inthe Z-X plane.

"General Solutions for Tailored Modulation Profiles in AdiabaticExcitation", by Thomas E. Skinner and Pierre-Marie L. Robitaille,published in the Journal of Magnetic Resonance 98, pp. 14-23 (1992),describes an inversion pulse having a triangular trajectory. FIG. 3shows an example of such a trajectory.

"Single-Shot, B1-Insensitive Slice Selection with a Gradient-ModulatedAdiabatic Pulse, BISS-8", by Robin A. de Graaf, Klaas Nicolay andMichael Garwood, published in Magnetic Resonance in Medicine 35:652-657(1996), describes a method for generating an optimal slice-selectionpulse, named BISS-8, having an adjustable flip angle. A main benefit ofthe BISS-8 pulse is that it does not scramble the phase of the selectedslice (which most adiabatic pulses do), so it can also be used for 180refocusing in spin-echo imaging. The BISS-8 pulse requires much moreamplitude than comparable pulses. However, the peak required amplitudeis lower than comparable pulses. In addition, both the gradients and theRF frequency are modified during a BISS-8 pulse.

SUMMARY OF THE INVENTION

It is an object of some embodiments of the present invention to providean adiabatic inversion pulse having a shorter duration than prior-artadiabatic pulses.

It is another object of some embodiments of the present invention toprovide an adiabatic inversion pulse having substantially independentlyselectable transition widths and bandwidths.

It is a further object of some embodiments of the present invention toprovide an analytically derived adiabatic pulse which is easily modifiedto meet different design constraints. In a preferred embodiment of thepresent invention, the adiabatic pulse is defined in a parametricmanner.

The inventors of the present invention have determined that an adiabaticpulse having a substantially rectangular trajectory can be made of ashorter duration than a pulse having a half-ellipse shaped trajectory,such as a sech/tanh pulse which is known in the art. A pulse inaccordance with a preferred embodiment of the invention includes ahorizontal segment, a vertical segment and a short curved segmentconnecting the vertical and horizontal segments. Preferably, both thetrajectory and its derivatives are continuous, to ensure tracking. Theterms rectangular pulse and half-rectangle trajectory pulse are usedinterchangeably in this specification and denote an adiabatic RF pulsehaving a substantially half-rectangle trajectory, with, preferably,rounded corners.

It should be appreciated that in a half-rectangle shaped trajectory, theradius (r) is always longer than (or equal to) that of a similarhalf-ellipse trajectory. It then follows directly from the adiabaticcondition (Γ=r/|θ|>>1) that for similar values of Γ, the rate of change(θ) can be made larger, and thus, the pulse duration, shorter. Moreover,the peak RF amplitude of required by the half-rectangle pulse is nogreater than that of a corresponding half-ellipse pulse.

In a preferred embodiment of the invention, the transition width (2c₀),slice width (SW) and duration (T) of a pulse are interrelated such thatany two of these parameters of the pulse can be varied substantiallyindependently of each other, with the pulse remaining adiabatic andefficient. Since the transition width is independently controllable fromthe bandwidth, a shorter inversion pulse can be obtained than ispossible using a standard pulse, such as a tanh/sech pulse, albeit atthe expense of increasing the transition width.

Another aspect of the present invention relates to an optimizationmethod, including, selecting a trajectory and then determining anoptimal rate of motion along the trajectory. Preferably, the trajectoryis selected by minimizing the integral ∫|dθ|/r to determine an optimaltrajectory.

In a preferred embodiment of the invention, the rate of motion along thetrajectory is determined responsive to a maximum rate of motion whichsatisfies the adiabatic condition for all the spins in the slice foreach point P along the trajectory.

Preferably, an optimal velocity profile is determined and then thedetermined maximum rate of motion is used to scale the velocity profile.Alternatively or additionally, the rate of motion along the trajectoryis numerically optimized.

In a preferred embodiment of the invention, the maximum rate of motionis determined for a mathematical support region which defines whichportions should be inverted (the slice) and which not (out-of-slice).Preferably, the support region also includes a range of expected localRF field strengths, such that the pulse can be verified as adiabatic forthe expected RF range. Typically, the support regions will berectangular (Larmor frequency range x RF amplitude range). In apreferred embodiment of the invention, the support regions arenon-rectangular.

In accordance with another preferred embodiment of the invention, aninversion pulse is obtained by projecting a rectangular trajectory pulseonto a different path shape, preferably, one intermediate a halfrectangle and a half-ellipse. In a preferred embodiment of theinvention, at least 50% of the trajectory is outside a trajectorydescribed by a half ellipse. More preferably, at least 70% is outsideand most preferably, at least 90% is outside. The percentages refer toan angular measure relative to the slice center, in which for eachangular unit of the trajectory, the trajectory is considered to beoutside a half-ellipse trajectory if the distance r is longer for thetrajectory than for the half-ellipse trajectory.

When two trajectories are compared to determine the above relationship,they should be scaled to have the same slicewidth and the same peak RFamplitude. Typically, the maximum available peak RF amplitude is used soas to minimize the pulse duration. However, it should be noted that ahalf-rectangular trajectory can achieve the same inversion as ahalf-ellipse trajectory having a similar duration, but using a lowerpeak RF amplitude.

Alternatively to comparing a trajectory in accordance with a preferredembodiment of the invention to a half-ellipse trajectory, a trajectoryof the invention may be compared to a parametric family of trajectories.A standard half-ellipse trajectory may be defined as a sin/costrajectory. The parametric family of trajectories is defined assin.sup.α /cos.sup.α where 0<α<1. The extreme case where α→0 is arectangular trajectory. All of these trajectories are "outside" ahalf-ellipse trajectory. A trajectory in accordance with a preferredembodiment of the invention is preferably either the same as or mostlyoutside (50%, 70% or 90%) a sin.sup.α /cos.sup.α trajectory. Preferably,α<0.9, more preferably, α<0.6, most preferably, α<0.4.

Another aspect of the present invention relates to magnetic resonanceimaging devices employing an inversion pulse determined using the aboveoptimization method.

Yet another aspect of the present invention relates to a method ofapplying an inversion pulse, including determining a desired inversionbandwidth, independently determining a desired transition width andapplying an RF pulse having a predetermined peak RF amplitude and aminimum duration which yields the desired bandwidth and transitionwidth.

A method of applying an inversion pulse in accordance with anotherpreferred embodiment of the invention, includes determining a desiredpulse duration and peak RF amplitude for the pulse, selecting a tradeoffbetween an inversion bandwidth and a transition width of the pulse andapplying an RF inversion pulse which yields the selected tradeoff.

It should be appreciated that a pulse generated in accordance with thepresent invention may also be used for flipping spins by angles otherthan 180 degrees, such as by 90 degrees, especially for on-resonanceexcitation or as a part of a composite pulse.

There is therefore provided in accordance with a preferred embodiment ofthe invention, a method of inverting spins for magnetic resonanceimaging, comprising:

subjecting the spins to a strong magnetic field; and

irradiating the spins with an adiabatic RF pulse having a trajectory,wherein said trajectory comprises, in a Z-X rotating frame of referencewhich rotates at the instantaneous frequency of the RF pulse:

a substantially horizontal segment;

a substantially vertical segment; and

a curved segment connecting the horizontal segment and the verticalsegment. Preferably, the trajectory further comprises a secondhorizontal segment and a second curved segment connecting the verticalsegment and the second horizontal segment.

Alternatively or additionally, the RF pulse is analytically described.Alternatively or additionally, an adiabatic parameter is maintained at aminimum value which ensures tracking for a predefined support region ofthe spins, for substantially the entire trajectory. Preferably, theminimum value is determined based on an expected range of RF fieldinhomegeneities at the spins.

There is also provided in accordance with a preferred embodiment of theinvention, a method of inverting spins for magnetic resonance imaging,comprising:

subjecting the spins to a strong magnetic field;

selecting a desired duration of an inversion pulse;

selecting a transition width of said pulse, which width is between 1.4and 1.9 divided by the duration and which is otherwise substantiallyindependent of said duration;

analytically generating an adiabatic RF pulse having the desiredduration and the desired transition width; and

inverting the spins with the generated RF pulse, by irradiating thespins with the pulse.

Preferably, the transition width is less than 1.7 divided by theduration. More preferably, the transition width is less than 1.5 dividedby the duration.

There is also provided in accordance with yet another preferredembodiment of the invention a method of inverting spins for magneticresonance imaging, comprising:

subjecting the spins to a strong magnetic field; and

irradiating the spins with an adiabatic RF pulse,

wherein, said adiabatic pulse is a projection of an adiabatic pulsehaving a substantially rectangular trajectory.

There is also provided in accordance with yet another preferredembodiment of the invention a method of inverting spins for magneticresonance imaging, comprising:

subjecting the spins to a strong magnetic field;

selecting a desired bandwidth to invert;

selecting a desired duration of an inversion pulse, which duration isless than 90% of the duration of a shortest possible sech/tanh pulsewhich achieves an inversion of the desired bandwidth under similar peakRF amplitude limitations; analytically generating an adiabatic RF pulsehaving the selected duration and selected bandwidth; and

inverting the spins with the generated RF pulse, by irradiating thespins with the pulse.

Preferably, said duration is less than 70% of the duration of thesech/tanh pulse. More preferably, said duration is less than 50% of theduration of the sech/tanh pulse.

There is also provided in accordance with yet another preferredembodiment of the invention, a method of inverting spins for magneticresonance imaging, comprising:

subjecting the spins to a strong magnetic field; and

irradiating the spins with an adiabatic RF pulse having a trajectory,wherein at least 50% of said trajectory, in a Z-X rotating frame ofreference which rotates at the instantaneous frequency of the RF pulse,is outside a trajectory defined by sin.sup.α /cos.sup.α, wherein, α<0.9.

Preferably, α<0.7. More preferably, α<0.5. Optionally, α<0.4.

Alternatively or additionally, at least 50% comprises at least 70%. Morepreferably, at least 50% comprises at least 90%.

There is also provided in accordance with another preferred embodimentof the invention, a method of generating an optimal inversion pulse,comprising:

defining at least one support region, including at least one in-sliceregion;

selecting a trajectory;

determining, for each point P along the trajectory, the maximum sweeprate which ensures tracking over the at least one support region; and

generating a velocity profile from the determined maximum sweep rates.

In a preferred embodiment of the invention, the velocity profile isanalytically defined. Alternatively or additionally, the trajectory is ahalf-ellipse trajectory. Alternatively, the trajectory is ahalf-rectangle trajectory.

Alternatively or additionally, generating a velocity profile comprisessearching for a minimum adiabatic parameter which ensures tracking overall the at least one support region.

In a preferred embodiment of the invention, the method includesoptimizing the velocity profile without taking into account theadiabatic condition.

Alternatively or additionally, the at least one support region comprisesat least two out-of slice support regions. Alternatively, the at leastone support region comprises only one out-of slice support region.

Alternatively or additionally, determining the maximum sweep ratecomprises finding a point in the support region having a minimumadiabatic parameter. Preferably, finding a point comprises analyticallyfinding a point, at least in one dimension of the support region.

Alternatively or additionally, said support regions have a first extentdetermined by the slice profile and a second extent determined by anexpected RF amplitude range. In a preferred embodiment of the invention,the support regions are non-rectangular.

Alternatively or additionally, the method comprises setting a distancebetween the support regions, responsive to a desired transition width.

There is also provided in accordance with a preferred embodiment of theinvention, a method of optimizing an adiabatic inversion pulse,comprising:

selecting a trajectory having a velocity profile; and

optimizing the velocity profile, along the trajectory and without usingthe adiabatic condition as a constraint,

wherein, the trajectory is a constraint of the optimization.

Preferably, optimizing along the trajectory comprises optimizing usingoptimal control methods. Alternatively or additionally, optimizing alongthe trajectory comprises constraining the velocity profile to allow onlyforward motion along the trajectory.

Alternatively or additionally, the trajectory is a substantiallyhalf-rectangular trajectory.

There is also provided in accordance with a preferred embodiment of theinvention, a method of inverting spins for magnetic resonance imaging,comprising, applying an inversion pulse produced by any of the abovedescribed pulse generation techniques

There is further provided in accordance with a preferred embodiment ofthe present invention, apparatus for magnetic resonance imagingutilizing an inversion method according to any of the inversion methodsdescribed above and/or which radiates an RF pulse generated according toany of the above described methods of generating RF pulses.

There is also provided in accordance with a preferred embodiment of theinvention an RF pulse generated according to any of the above describedmethods of generating an RF pulse.

There is provided in accordance with a preferred embodiment of theinvention, an adiabatic RF inversion pulse having a trajectory, whichwhen described in a Z-X rotating frame of reference which rotates at theinstantaneous frequency of the RF pulse, comprises:

a substantially horizontal segment;

a substantially vertical segment; and

a curved segment connecting the horizontal segment and the verticalsegment.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be more clearly understood by the following detaileddescription of preferred embodiments of the invention, taken togetherwith the drawings in which:

FIG. 1 is a graph showing a typical prior art (inversion) slice profile;

FIG. 2 is a graph showing a trajectory of a typical prior art adiabaticpulse;

FIG. 3 is a graph of a prior art triangular trajectory pulse;

FIG. 4 is a graph of a trajectory of a preferred prior art,substantially rectangular inversion pulse as described herein;

FIG. 5 is a graph illustrating the adiabatic performance of arectangular trajectory pulse in accordance with a preferred embodimentof the invention;

FIG. 6 is a graph illustrating the efficiency of a rectangulartrajectory pulse in accordance with a preferred embodiment of theinvention;

FIGS. 7A and 7B are illustrations of support regions for generatingvarious optimal pulses, in accordance with preferred embodiments of theinvention; and

FIG. 8 is a schematic drawing of a projection of a rectangulartrajectory pulse in accordance with a preferred embodiment of theinvention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

One aspect of the present invention relates to a method of optimizing aninversion pulse. The method includes two steps:

(a) selecting a trajectory; and

(b) selecting a fastest velocity profile for the trajectory, which willstill ensure inversion in the slice. Preferably, the fastest profile isdetermined by solving the Bloch equations for different trial velocityprofiles and selecting the fastest velocity profile which still ensuresinversion. In addition, in order to ensure tracking, both the trajectoryand its derivatives are preferably constrained to be continuous.Preferably, at least the first derivative is continuous.

FIG. 4 is a graph of an optimal trajectory in accordance with apreferred embodiment of the invention. The trajectory is preferablysymmetrical and each half preferably comprises a substantiallyhorizontal segment AC, a substantially vertical segment DF and a curvedsegment CD bridging the horizontal segment and the vertical segment.U.S. patent application Ser. No. 08/916,390, "Frequency SelectivePulse", filed Aug. 23, 1997 by applicant Elscint LTD., the disclosure ofwhich is incorporated herein by reference "Design of Adiabatic Pulsesfor Fat-Suppression Using Analytic Solutions of the Bloch Equation", byD. Rosenfeld, S. L. Panfil and Y. Zur, in Magnetic Resonance in Medicine37:793-801 (1997) and "Analytic Solutions of the Bloch EquationInvolving Asymmetric Amplitude and Frequency Modulations", by D.Rosenfeld, S. L. Panfil and Y. Zur, in Physical Review A, Vol. 54, pp.2439-2443 (1996), the disclosures of which describe the use ofnon-symmetrical pulses, especially for selectively inverting fat tissue."A New Adiabatic Inversion Pulse", by Daniel Rosenfeld and Yuval Zur,published in Magnetic Resonance in Medicine, 36:124-136 (1996), thedisclosure of which includes mathematical analyses of the trajectoryshown in FIG. 4.

Referring back to FIG. 2, each group of spins along the Z axisexperiences a different effective magnetization vector as a result ofthe RF inversion pulse. For an arbitrary spin having a Larmor frequencyω₀ and an arbitrary point P along the trajectory, the effective magneticfield has a magnitude of r and a direction θ. The adiabatic parameter isgiven by Γ=r/|θ|. Γ obtains a minimal value with respect to an arbitrarypoint y^(P) _(min) along the Z axis (which may be other than ω₀). If atpoint y^(P) _(min) the adiabatic condition is fulfilled (for point P ofthe trajectory) it will also be fulfilled for all other points (of thein-slice region) along the Z axis, since, by definition, Γ will belarger for those points. Thus, by ensuring tracking for spins at y^(P)_(min) for every point P of the trajectory, tracking is established forall the spins in the slice.

Thus, in a preferred embodiment of the invention, tracking for eachpoint P is assured by setting Γ(y=y^(P) _(min)) to a value γ₀ which issufficient to ensure adiabatic inversion. The value of γ₀ depends onwhether the pulse parameters and the modulation functions are expressedin frequency units (Hz) or in units of angular frequency (rad/s). Whenexpressed in frequency units (as in the instant application), γ₀ issmaller by a factor of 2π than when expressed in units of angularfrequency. For this reason, γ₀ is shown as smaller than 1, even thoughthis would appear to violate the adiabatic condition. If γ₀ isincreased, the tracking is improved, but the pulse duration is extended.It will therefore be appreciated that a minimal value for γ₀ isdesirable. In accordance with a preferred embodiment of the invention, agiven trajectory/velocity profile pair is optimized by multiplying theentire velocity profile by a constant such that γ₀ is a minimal valuewhich ensures tracking. γ₀ =r/|dθ| (where r and dθ defined by P andy^(P) _(min), so the pulse duration may be written as ##EQU1##

In a preferred embodiment of the invention, a constant value of γ₀ isselected for the entire trajectory. The value of y^(P) _(min) isdetermined analytically and the integral ∫|dθ|/r is minimized todetermine the trajectory. Thereafter, a minimum value of γ₀ whichensures inversion, is selected, i.e., γ₀ is a time scale constant.Alternatively, a different value of γ₀ may be selected for each portionof the trajectory. Thus, a fast trajectory is combined with a fastvelocity profile to yield a short duration pulse.

Referring back to FIG. 4, a pulse in accordance with a preferredembodiment of the invention may be analytically defined as comprising ofdifferent segments, AB, BC, CD and DF. The following equations werederived analytically by minimizing the above integral and they describeboth the trajectory and the general velocity profile for the trajectory.For convenience, a parameter t in the equations is assumed to be zero atthe start of each segment. Segment AB is a horizontal segment of thetrajectory which has a constant adiabatic parameter with respect to apoint Q, which is on the border between the in-slice region and thetransition region: ##EQU2##

    z.sub.AB (t)=SW/2                                          (2)

Point B, having an X coordinate of x₀, is defined as the point where thespin with y^(P) _(min) enters the in-slice region, whereby

    x.sub.0 =√2c.sub.0                                  (3)

The duration of segment AB is ##EQU3##

Section BC is a horizontal segment where y^(P) _(min) is in the in-sliceregion, where ##EQU4##

    z.sub.BC (t)=SW/2                                          (6)

Point C is the start of the curved segment and starts at ##EQU5## whereX_(f) is the peak RF amplitude. The duration of segment BC is ##EQU6##

Segment CD is a curved segment which connects the horizontal portion ofthe trajectory at C and the vertical portion of the trajectory at D.

    x.sub.CD (t)=(1+2.63τ-2.87τ.sup.2 -8.04τ.sup.3)x.sub.1(9)

    z.sub.CD (t)=SW/2+(0.064τ-7.62τ.sup.2 +9.90τ.sup.3)x.sub.1(10)

where τ is a dimensionless variable defined by ##EQU7##

The duration of segment CD is ##EQU8## and ZD is

    z.sub.D =SW/2-0.276x.sub.1                                 (13)

Segment DF is a vertical segment, where

    x.sub.DF (t)=x.sub.f                                       (14) ##EQU9##

The duration of segment DF is ##EQU10##

Preferably, the maximum available X_(f) is be used, as long as ##EQU11##

The total duration of the pulse is twice the sum of the durations of theindividual segments: ##EQU12##

It should be appreciated that the above described pulse is analyticallyand parameticly described and as such more amenable to adaptation todifferent MRI devices and imaging sequences. In particular, analyticalpulses are easier to implement and adapt to a particular RF synthesizerthan numerically defined pulses. The inventors have determined that theminimal value of γ₀ which renders satisfactory inversion isapproximately 4/2π. However, as described below, lower values of γ₀ arepossible for at least portions of the trajectory.

Referring to Equation (18), a first unique property of a rectangulartrajectory pulse such as that shown in FIG. 4 is that the slice widthand the transition width may be individually controlled. Once a pulseduration and peak RF amplitude (x_(f)) are set the transition width(2c₀) and the slice width (SW) may be traded off, to a limit determinedby equation (18). The segment AB of the trajectory controls transitionwidth, while the segment BF controls the slice width. If, for a givenpulse duration, the time along segment AB is decreased and the timealong segment BF is increased, then the quality of tracking in thetransition region will be reduced and the transition width willincrease. Conversely, the quality of tracking in the in-slice regionwill increase, thereby ensuring inversion. Thus, in a very short pulse,the transition width may be sacrificed to assure inversion. Inparticular, shorter pulse durations than the prior art can be achievedat the expense of transition width, which in many cases is not asimportant a consideration as slice width or pulse duration, for example,as described with reference to FIG. 6, below. In addition, when the RFpeak amplitude is limited, transition width and/or slice width and peakRF amplitude can also be traded-off (preferably, according to equation18).

In an extreme example, to invert the largest possible bandwidth in agiven duration and using a given maximum RF amplitude, the transitionwidth is sacrificed (allowed to increase). Since most of the pulse willbe spent in the vertical portions of the trajectory, the first term ofequation (18) should be minimized. This occurs when x₀ is equal to x₁(see equation (8)). The resulting transition width is then c₀=0.51x_(f). Also, c₀ <=SW/2, since the transition region cannot exceedthe inverted bandwidth. By substituting the transition width back intoequation (18), ##EQU13## which is an estimate of the maximum invertablebandwidth.

In the other extreme, to achieve a narrowest possible transition, thebandwidth may be sacrificed. Since most of the time is spent in thetransition region AB, the first term of equation (18) will dominate thesecond term: ##EQU14##

It should be appreciated that in many cases the pulse duration is animportant limitation, such as in T₂ imaging, where an RF pulse durationshould be much shorter than the T₂ decay time. In some imagingsequences, using a shorter pulses significantly increases the efficiencyof data collection. For example, in an inversion-recovery MRI sequence,using a shorter inversion pulse allows more image slices to besimultaneously imaged than by using a sech/tanh pulse.

FIG. 5 is a graph illustrating that a rectangular trajectory pulse inaccordance with a preferred embodiment of the present invention requiresa lower RF amplitude that a comparable sech/tanh pulse. The pulseduration is 9 ms, the slice width SW/2π is 8 kHz and the value of γ₀ is3.4/2π. FIG. 5 plots the Z component of the final magnetization at theslice center as a function of the maximal RF amplitude (γB₁ /2π, which,as described above, is in units of frequency). Full inversion can beachieved at as low a field strength as 0.8 kHz for a rectangulartrajectory as compared to 1.5 kHz for a sech/tanh pulse.

FIG. 6 is a graph illustrating the efficiency of a rectangulartrajectory pulse in accordance with a preferred embodiment of theinvention. For a slice width of 8 kHz and a maximum RF amplitude of 0.8kHz, a rectangular trajectory pulse can achieve complete inversion (atthe slice center) in less than one third the duration of an equivalentsech/tanh pulse.

It should be appreciated that as the strength of the static magneticfield increases, the absorption of RF by the body increasesdramatically. This has the effect of reducing the maximum available RFamplitude (more is absorbed). In addition, the increase in magneticloading of the patients body causes the RF to become more inhomogeneous.The specific absorption rate (SAR) of a pulse is an indication of theamount of RF energy absorbed by the subject's body. The RF energy of apulse of a duration T is proportional to ##EQU15##

The SAR of the pulse described with reference to FIG. 4 is (only) 1.18times the SAR of a sech/tanh pulse, having a similar bandwidth,transition width and peak RF amplitude, but having a longer durationthan the pulse shown with relation to FIG. 4.

Another aspect of the present invention relates to numericallyoptimizing the velocity profile of an adiabatic pulse without regard tothe adiabatic condition. The use of a constant γ₀ for the entiretrajectory may induce imperfections in the selected slice, in the formof sidelobes in the out of slice-region or ripples in the invertedregion. In a preferred embodiment of the invention, instead of using aconstant value of γ₀ for the entire trajectory, γ₀ is optimized alongthe trajectory. MRI pulse optimization is well known and the optimalityis usually measured with respect to a cost functional which expresses adistance between the target magnetization m_(d) (ω₀) and the actualmagnetization m(ω₀,T), e.g., ##EQU16## where the sum is over a range ofLarmor frequencies, including the region of inversion. D. Rosenfeld andY. Zur, Magnetic Resonance in Medicine, Vol. 36, p 401, (1996),describes an optimization method in which both amplitude and frequencymodulation functions are optimized. The adiabaticity is incorporatedinto the optimization by enhancing the functional of equation (22) withan additional, adiabaticity-preserving term, the purpose of which is tomaintain adiabaticity during the optimization process.

In a preferred embodiment of the present invention a functional is addedto equation (22) which does not contain any adiabaticity-relatedportions. Only the rate of motion along the trajectory is optimized. Itis assumed that adiabaticity is preserved by the following tworestrictions: the trajectory is unchanged by the optimization and onlyforward motion (of the effective vector) along the trajectory isallowed. The optimization may also be performed to provide a pudse witha narrow transition region (suitable for fat suppression), to correctfor defects which occur when a parametric pulse is taken to extremevalues of its parameters (or beyond where it is supposed to beadiabatic) .

Another aspect of the present invention relates to an expansion of theabove described procedure for deriving an optimal velocity profile forthe rectangular trajectory to a procedure for generating efficientpulses having rectangular or other trajectories. The pulses are tailormade for a given slice-inversion situation. In accordance with apreferred embodiment of the invention the pulse is designed to beadiabatic for a predetermined range of RF field strengths which areexpected inside the patient's body. Thus, reducing the peak RF amplitudeto that which is required to maintain adiabatic behavior in the regionshaving the lowest RF amplitude. One reason why adiabatic pulses arepreferred is because an adiabatic pulse works well even when the RFfield is not homogeneous. A typical adiabatic pulse will invert twospins equally well even if there is a factor of two between their localRF field strength. Briefly, increasing the available RF amplitudeincreases r more than it increases θ, so the adiabatic condition ismaintained. Nevertheless, there are limits to the variability in fieldstrength which can be accommodated for by an adiabatic pulse. Inaddition, as described above, the minimum amplitude which will ensureinversion is desired. In accordance with a preferred embodiment of theinvention, if the range of effective magnetic field strengths is knownin advance, it is possible to tune the pulse to these expected fieldstrengths.

FIG. 7A illustrates a support region 10, corresponding to an in-sliceregion and support regions 12 and 14 corresponding to out-of-sliceregions, all regions defined for a particular inversion pulse. In manycases, only the in-slice support region is required to determine anoptimal velocity profile. The horizontal axis, Ω₀, is a Larmor frequencyaxis. As shown in FIG. 7A, between Larmor frequencies -SW/2+c₀ andSW/2-c₀ the spins must be inverted by the pulse; frequencies under-SW/2-c₀ or over SW/2+c₀ must not be inverted by the pulse; and otherfrequencies, by definition, fall into the transition region, wherebythey are partially inverted. The vertical axis, γ^(B) _(max), is the RFamplitude axis. The vertical shaded extent of the support regionsindicates the range of RF amplitudes for which the pulse is required tobe adiabatic. In accordance with a velocity profile generating methodaccording to a preferred embodiment of the present invention, atrajectory is selected and an optimal velocity profile along thetrajectory is found such that the adiabatic condition be maintained forall points in the three support regions. Typically, an optimal velocityprofile is one which is faster. Alternatively, a more optimal velocityprofile can use a trajectory which requires a lower peak RF amplitudeand achieves the same inversion in a similar period of time.

After the support regions are defined, a point y^(P) _(min) (as definedabove) is determined for each point P along the trajectory, under theconstraint that the point y^(P) _(min) must be inside one of the supportregions (usually y^(P) _(min) falls within the in-slice support region.The value of Γ at y^(P) _(min) is set (thereby setting the angularvelocity at point P) to a constant γ₀, which is basically a time-scalingconstant. A velocity profile for the trajectory is generated from therequired velocity at each point P. The magnitude of γ₀ is preferablydetermined by searching for a value of γ₀ which ensures a satisfactorymagnetization profile. The magnetization inversion is calculated bysolving the Bloch equations (simulation) for different trial values ofγ₀. The minimal γ₀ that can still render satisfactory inversion isfinally selected. The search is preferably performed using a binarysearch method. In some cases this value of γ₀ might not ensure completeinversion at y^(P) _(min).

y^(P) _(min) may be found numerically. Once γ₀ is found, the velocityprofile may be generated (numerically) from the values of Γ at eachy^(P) _(min). However, in a preferred embodiment of the invention,described below, y^(P) _(min) and/or the velocity profile are foundusing analytical methods, thereby requiring fewer computations andproviding greater flexibility.

In a preferred embodiment of the invention the velocity profile isgenerated by integrating the following differential equation: ##EQU17##and then inverting the resulting function t(η) to yield η(t). η(t) is amonotonic ascending function of t which varies from -π/2 to +π/2 anddefines the velocity profile along the trajectory. For example, thesech/tanh pulse may be parametrized by:

    x(t)=Acos(η(t))

    z(t)=ω.sub.c -Bsin(η(t))                         (24)

where A is the RF peak amplitude, B is SW/2 and ω_(c) is the Larmorfrequency at the slice center. Each specific value of η represents isassociated with a specific point along the trajectory. f_(m) (η), whichis related to Γ, is a shorthand for the minimum value of a functionf(η,Ω₀,v) for a given value of η over a range of RF inhomogenities v anda range of Larmor frequencies Ω₀ . This function f is defined in thefollowing manner. Referring back to FIG. 2, for each point P along thetrajectory and an arbitrary spin, r and θ are a function of x and y. TheRF amplitude, x, is the applied RF amplitude ω₁, scaled by v. Inaddition, both x and y are a function of η(t). Thus:

    x(t)=νω.sub.1 (η(t))

    y(t)=Δω(Ω.sub.0,η(t))                (25)

Thus, r and θ are (using a Cartesian to polar transformation): ##EQU18##The adiabatic parameter can be rewritten using equations (25) and (26)as: ##EQU19## where g'=dg/dη and g=dg/dt; recalling tha dg(η(t))|dt=g'η.

Since, by definition, Γ≧γ₀, by reordering equation (27): ##EQU20##thereby defining f. f_(m) (η) is the minimum of f for a particular valueof η (a specific point along the trajectory) over the support region. Itmay be expected that different values of η will yield different minimumvalues of f_(m). As a result of the relationship between η(t) and f_(m)(η) in equation (28), dη(t)/dt is shown to be a function of η (and nott). Thus, t(η) can be found from equation (23).

In some preferred embodiments of the invention y^(P) _(min) is found bysearching over the support regions and not by analytical derivation,since the support regions are usually irregular in shape. In the priorart it has usually (if not always) been assumed that that y^(P) _(min)occurs at the synthesizer frequency of the point P. However, this isusually not the case. A preferred search method uses an analyticallydefined function y^(P) _(min) to reduce the computational complexity offinding f_(m).

An analytic expression for y^(P) _(min), (the off-resonance frequency atwhich f_(m) is minimized) for a single RF amplitude is: ##EQU21## where,ν is the absolute value of the slope of the trajectory at point P andwhere x is the x position (RF amplitude) of point P. It should beappreciated that y^(P) _(min) is an implicit function of the peak RFamplitude, since changing the RF amplitude affects the slope, ν, at eachpoint P of the trajectory. Preferably, the range of RF amplitudes isdivided into subsections. For each subsection, y^(P) _(min) isdetermined. If y^(P) _(min) fall outside the support regions, the twopoints in the neighboring support regions which are closest to y^(P)_(min) are selected and the smaller value of f at the two points isused. Then, f_(m) is determined by selecting the lowest value of f forall of the subsections. In an alternative preferred embodiment of theinvention, f_(m) is determined using a one-dimensional search technique,as known in the art, on the range of RF amplitudes. The one-dimensionalsearch need only evaluate f, not its derivatives.

FIG. 7B illustrates a design of support regions for a fat suppressionpulse. A special requirement of fat suppression pulses is that theentire transition region fit in the small frequency difference betweenfat and water. In a preferred embodiment of the invention, thisrequirement is met by designing support regions, whose only constraintsare the width of the transitional region between an in-slice region 20and an out-of-slice region 22 and that the support regions cover thespectral bandwidth of water and fat, respectively. In FIG. 7B the widthconstraint is the width 2c₀. Of course, the desired range of RFamplitudes should also be defined. It should be noted that in FIG. 7Bonly one out-of-slice region is defined, since the most significantrequirement in fat suppression is that the fat be inverted and the waternot. There is generally no interest in the what happens at frequencieslower than the fat frequencies. In cases where there is such aninterest, an addition out-of-slice support region may be defined. Thefat suppression pulse is then determined in a manner similar to thatdescribed with respect to FIG. 7A.

A fat suppression pulse determined in this manner is more efficient thana comparable sech/tanh pulse, since only a narrow bandwidth is required,and the above described bandwidth/transition-width/RF amplitude tradeoffmay be applied. Further, since only one transition region is important,the behavior of the pulse after that narrow transition is obtained ismuch less constrained. As a result, fat suppression pulses can have aportion which follows a half-ellipse trajectory, with an optimalvelocity profile while the rest of the pulse can have a differenttrajectory, such as a straight line.

It should be appreciated that the present invention is not limited torectangular trajectory pulses as shown in FIG. 4. Rather, the inventorshave determined that the properties of the pulse improve as thetrajectory approaches the form of a rectangle. Thus, a less optimal, butstill useful, pulse may be obtained by projecting the pulse onto adifferent trajectory, such as one intermediate a rectangle and anellipse. Projection is described in "Variable Rate SelectiveExcitation", by Steven Conolly, Dwight Nishimura and Albert Macovski, inJournal of Magnetic Resonance, Vol. 78, pp. 440-458 (1988). FIG. 8 showsan example of such a projected trajectory which is intermediate atrajectory as shown in FIG. 4, in accordance with the present invention,and an elliptical trajectory, as show in FIG. 2. The trajectories ofFIGS. 2 and 4 are shown as dotted lines. In addition, in accordance withother embodiment of the present invention, the velocity profileoptimization technique may be applied to any trajectory. Alternatively,the rectangular trajectory may be used with other, possibly lessoptimal, velocity profiles. It should, however, be appreciated, thatprojecting the trajectory decreases the parameter "r" by a certainfactor for each point along the trajectory, thus, the duration T istypically extended by that factor, at each point.

A parametric family of trajectories, which are intermediate ahalf-ellipse and a half-rectangle, can be generated, such that they arefaster than a half-ellipse trajectory. A standard half-ellipsetrajectory may be defined as a sin/cos trajectory, for example as shownin equation (24). The parametric family of trajectories is defined as

    x(t)=Acos.sup.α (η(t))

    z(t)=ω.sub.c -Bsin.sup.α(η(t))             (30)

referred to herein as sin.sup.α /cos.sup.α where 0<α<1. The extreme casewhere α→0 is a half-rectangular trajectory. All of these trajectoriesare "outside" a half-ellipse trajectory. Preferably, α<0.9, morepreferably, α<0.6, most preferably, α<0.4.

In addition, it should be appreciated that the above describedoptimization methods are applicable to any trajectory, not onlyrectangular trajectories and can also both be sequentially applied.However, since many optimization methods are sensitive to the startingpoint, it is useful to start with a more efficient trajectory, i.e., ahalf-rectangular trajectory.

It will be appreciated by a person skilled in the art that the presentinvention is not limited by what has thus far been described. Rather,the scope of the invention is limited only by the claims which follow.

What is claimed is:
 1. A method of inverting spins for magneticresonance imaging, comprising:subjecting the spins to a strong magneticfield; selecting a desired duration of an inversion pulse; substantiallyindependently of said duration, selecting a transition width of saidpulse, such that the width is between 1.4 and 1.9 divided by theduration; analytically generating an adiabatic RF pulse having thedesired duration and the desired transition width; and inverting thespins with the generated RF pulse, by irradiating the spins with thepulse.
 2. A method according to claim 1, wherein said transition widthis less than 1.7 divided the duration.
 3. A method according to claim 1,wherein said transition width is less than 1.5 divided the duration. 4.A method according to claim 1, wherein analytically generating a pulsecomprises analytically generating a pulse having a non-constant Rfamplitude.
 5. A method of inverting spins for magnetic resonanceimaging, comprising:subjecting the spins to a strong magnetic field; andirradiating the spins with an amplitude modulated adiabatic RF pulsehaving a trajectory, wherein at least 50% of said trajectory, in a Z-Xrotating frame of reference which rotates at the instantaneous frequencyof the RF pulse, is outside a trajectory defined by sin.sup.α/cos.sup.α, wherein, α<0.9.
 6. A method according to claim 5, whereinsaid trajectory starts and ends with a substantially zero RF amplitude.7. A method according to claim 5, wherein α<0.7.
 8. A method accordingto claim 5, wherein α<0.5.
 9. A method according to claim 5, whereinα<0.4.
 10. A method according to claim 5, wherein at least 50% comprisesat least 70%.
 11. A method according to claim 5, wherein at least 50%comprises at least 90%.
 12. A method of generating an optimal inversionpulse, comprising:defining at least one support region, including atleast one in-slice region; selecting a trajectory; determining, for eachpoint P along the trajectory, the maximum sweep rate which ensurestracking over the at least one support region; and generating a velocityprofile from the determined maximum sweep rates.
 13. A method accordingto claim 12, wherein the velocity profile is analytically defined.
 14. Amethod according to claim 12, wherein the trajectory is a half-ellipsetrajectory.
 15. A method according to claim 12, wherein the trajectoryis a half-rectangle trajectory.
 16. A method according to claim 12,wherein generating a velocity profile comprises searching for a minimumadiabatic parameter which ensures tracking over all the at least onesupport region.
 17. A method according to claim 12, comprisingoptimizing the velocity profile without taking into account theadiabatic condition.
 18. A method according to claim 12, wherein said atleast one support region comprises at least two out-of slice supportregions.
 19. A method according to claim 12, wherein said at least onesupport region comprises only one out-of slice support region.
 20. Amethod according to claim 12, wherein determining the maximum sweep ratecomprises finding a point in the support region having a minimumadiabatic parameter.
 21. A method according to claim 20, wherein findinga point comprises analytically finding a point, at least in onedimension of the support region.
 22. A method according to claim 12,wherein said support regions have a first extent determined by the sliceprofile and a second extent determined by an expected RF amplituderange.
 23. A method according to claim 22, wherein said first extentrepresents a nod-zero slice width.
 24. A method according to claim 22,wherein the support regions are non-rectangular.
 25. A method accordingto claim 12, comprising setting a distance between the support regions,responsive to a desired transition width.
 26. A method of optimizing anadiabatic inversion pulse, comprising:selecting a trajectory having avelocity profile; and optimizing the velocity profile, along thetrajectory without using the adiabatic condition as a constraint,wherein, the trajectory is a constraint of the optimization.
 27. Amethod according to claim 26, wherein optimizing along the trajectorycomprises optimizing using optimal control methods.
 28. A methodaccording to claim 27, wherein optimizing along the trajectory comprisesconstraining the velocity profile to allow only forward motion along thetrajectory.
 29. A method according to claim 28, wherein the trajectoryis a substantially half-rectangular trajectory.
 30. A method ofinverting spins for magnetic resonance imaging, comprising, applying aninversion pulse produced by the method of claim
 12. 31. A method ofinverting spins for magnetic resonance imaging, comprising, applying aninversion pulse produced by the method of claim
 26. 32. Apparatus formagnetic resonance imaging configured to perform an inversion methodaccording to claim
 1. 33. Apparatus for magnetic resonance imagingconfigured to perform an inversion method according to claim
 13. 34. AnRF pulse generated according to the method of claim
 12. 35. An RF pulsegenerated according to the method of claim 26.